The object of the paper are the so-called “unimaginable numbers”. In particular, we deal with some arithmetic and computational aspects of the Knuth’s powers notation and move some first steps into the investigation of their density. Many authors adopt the convention that unimaginable numbers start immediately after 1 googol which is equal to, and G.R. Blakley and I. Borosh have calculated that there are exactly 58 integers between 1 and 1 googol having a nontrivial “kratic representation”, i.e., are expressible nontrivially as Knuth’s powers. In this paper we extend their computations obtaining, for example, that there are exactly 2 893 numbers smaller than with a nontrivial kratic representation, and we, moreover, investigate the behavior of some functions, called krata, obtained by fixing at most two arguments in the Knuth’s power.
CITATION STYLE
Caldarola, F., d’Atri, G., Mercuri, P., & Talamanca, V. (2020). On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 11973 LNCS, pp. 381–388). Springer. https://doi.org/10.1007/978-3-030-39081-5_33
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